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The Noether-Lefschetz theorem and sums of 4 squares in the rational function field \(\mathbb{R}(x,y)\). (English) Zbl 0774.12002

It is well known that there exist polynomials in \(\mathbb{R}[x,y]\) which are positive on \(\mathbb{R}^ 2\) but not the sum of 3 squares in \(\mathbb{R}(x,y)\) [J. W. S. Cassels, W. J. Ellison and A. Pfister, J. Number Theory 3, 125-149 (1971; Zbl 0217.043)]. The author shows the existence of a whole family of polynomials in \(\mathbb{R}[x,y]\) which are positive on \(\mathbb{R}^ 2\), and thus the sum of four squares in \(\mathbb{R}(x,y)\), but not be sum of 3 squares in \(\mathbb{R}(x,y)\). He uses complex algebraic geometry in his proof, in particular the Noether- Lefschetz theorem [S. Lefschetz, Trans. Am. Math. Soc. 22, 327-482 (1921; JFM 48.0428.03)].

MSC:

12D15 Fields related with sums of squares (formally real fields, Pythagorean fields, etc.)
11E81 Algebraic theory of quadratic forms; Witt groups and rings
14P05 Real algebraic sets
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References:

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